Numismatically speaking, the exact difference between the two terms is the method of manufacture. Here’s a better example for us to understand: The Manila 1828, 1829 and the 1830 coins are minted by means of a screw press machine by “stamping” it unto a host coin either to obliterate or eliminate the insurgent legends. This term is called “counterstamp” or overstrike. In other countries such as Brazil, the same method was used but different purpose. The Brazilian government lack the neccessary local coinage to used in commerce so they recycle other nation’s coins by stamping their own government seal and design on the coins.
Now lets go to the term “countermark”. This method of manufacture is rather simple and small. This was made of a small circular hammer die also called as “golpe de martillo” in Spanish. Usually the host coin was placed in an anvil or flat surface. Then a small circular die with a handle was hardly punch onto the coin thus leaving a mark into it. This crude method was also used by the ancient Greek and Roman times.
This two terms must not used erronously when we try to describe a certain coin in order not to confused the description and to better understand the proper way of using these important numismatic terms.